Questions: Consider the rational function f(x) = (x^2 - 8x) / (x^2 - 2x - 48) Determine the discontinuities and the type of discontinuity, i.e. 'removable' or 'infinite'.

Solution Steps

To determine the discontinuities of the rational function, we need to factor the denominator and find the values of \( x \) that make the denominator zero. These values are the points of discontinuity. We then check if the numerator also becomes zero at these points to determine if the discontinuity is removable or infinite. If the numerator is zero at the same point, the discontinuity is removable; otherwise, it is infinite.

The given rational function is

\[ f(x) = \frac{x^2 - 8x}{x^2 - 2x - 48} \]

To find the discontinuities, we first factor the denominator:

\[ x^2 - 2x - 48 = (x - 8)(x + 6) \]

The discontinuities occur where the denominator is zero. Solving

\[ (x - 8)(x + 6) = 0 \]

gives the solutions \( x = 8 \) and \( x = -6 \).

To determine the type of discontinuity, we check if the numerator is also zero at these points.

• For \( x = 8 \), the numerator is \( 8^2 - 8 \times 8 = 0 \). However, since the denominator is also zero, the discontinuity is infinite.
• For \( x = -6 \), the numerator is \((-6)^2 - 8 \times (-6) = 84\), which is not zero. Therefore, the discontinuity is infinite.

Final Answer